An extension of Rauch comparison theorem to glued Riemannian spaces (Q1861077)

Let \(B_i\) be submanifolds of Riemannian manifolds \(M_i\) for \(i=1,2\). One supposes that \(B_1\) and \(B_2\) are isometric and identifies \(M_1\) with \(M_2\) along the subspaces \(B_i\) to define a glued Riemannian space. A curve on such a space is said to be a \(B\) geodesic if it is a geodesic in each manifold separately and if it satisfies a certain passage law. The notion of a \(B\)-Jacobi vector field is defined similarly. The author generalizes the Rauch comparison theorem to this context; the shape operators of the submanifolds \(B_i\subset M_i\) play a crucial role in the analysis.